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On the relative extrema of the Jacobi polynomials P 0 (0,-1) (x). (English) Zbl 0805.33014
Let ν k,n , k=1,,n; n=1,2, be the successive relative extrema of P n (0,-1) (x)/P n (0,-1) (1) when x decreases from +1 to -1. With a view to proving the conjecture of Askey that |ν k,n |<|ν k,n+1 |, first the authors derive some asymptotic approximations for the Jacobi polynomials P n-1 (0,1) (cosθ) and P n-1 (1,0) (cosθ), as n tends to , which lead them to obtain corresponding approximations for the zeros of P n-1 (1,0) (cosθ). These asymptotic approximations are then used to prove the conjecture stated above.
33C45Orthogonal polynomials and functions of hypergeometric type
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)